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Handbook of Mathematical Methods in Imaging pp 913–957Cite as

Statistical Methods in Imaging

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Abstract

The theme of this chapter is statistical methods in imaging, with a marked emphasis on the Bayesian perspective. The application of statistical notions and techniques in imaging requires that images and the available data are redefined in terms of random variables, the genesis and interpretation of randomness playing a major role in deciding whether the approach will be along frequentist or Bayesian guidelines. The discussion on image formation from indirect information, which may come from non-imaging modalities, is coupled with an overview of how statistics can be used to overcome the hurdles posed by the inherent ill-posedness of the problem. The statistical counterpart to classical inverse problems and regularization approaches to contain the potentially disastrous effects of ill-posedness is the extraction and implementation of complementary information in imaging algorithms. The difficulty in expressing quantitative and uncertain notions about the imaging problem at hand in qualitative terms, which is a major challenge in a deterministic context, can be more easily overcome once the problem is expressed in probabilistic terms. An outline of how to translate some typical qualitative traits into a format which can be utilized by statistical imaging algorithms is presented. In line with the Bayesian paradigm favored in this chapter, basic principles for the construction of priors and likelihoods are presented, together with a discussion of numerous computational statistics algorithms, including Maximum Likelihood estimators, Maximum A Posteriori and Conditional Mean estimators, Expectation Maximization, Markov chain Monte Carlo, and hierarchical Bayesian models. Rather than aiming to be a comprehensive survey, the present chapter hopes to convey a wide and opinionated overview of statistical methods in imaging.

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References and Further Reading

  1. Arridge SR, Kaipio JP, Kolehmainen V, Schweiger M, Somersalo E, Tarvainen T, Vauhkonen M (2006) Approximation errors and model reduction with an application in optical diffusion tomography. Inverse Probl 22:175–195

    Article  MathSciNet  MATH  Google Scholar 

  2. Bardsley J, Vogel CR (2004) A nonnegatively constrained convex programming method for image reconstruction. SIAM J Sci Comput 25:1326–1343

    Article  MathSciNet  Google Scholar 

  3. Bernardo J (2000) Bayesian theory. Wiley, Chichester

    MATH  Google Scholar 

  4. Bertero M, Boccacci P (1998) Introduction to inverse problems in imaging. IOP, Bristol

    Book  MATH  Google Scholar 

  5. Besag J (1974) Spatial interaction and the statistical analysis of lattice systems. J Stat Roy Soc 36:192-236

    MathSciNet  MATH  Google Scholar 

  6. Besag J (1986) On the statistical analysis of dirty pictures. J Roy Stat Soc B 48:259–302

    MathSciNet  MATH  Google Scholar 

  7. Besag J, Green P (1993) Spatial statistics and Bayesian computation, J Roy Stat Soc B 55:25–37

    MathSciNet  MATH  Google Scholar 

  8. Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New York

    MATH  Google Scholar 

  9. Björck Å (1996) Numerical methods for least squares problems. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  10. Boyles RA (1983) On the convergence of the EM algorithm. J Roy Stat Soc B 45:47–50

    MathSciNet  MATH  Google Scholar 

  11. Bruckstein AM, Donoho DL, Elad M (2009) From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev 51:34–81

    Article  MathSciNet  MATH  Google Scholar 

  12. Calvetti D (2007) Preconditioned iterative methods for linear discrete ill-posed problems from a Bayesian inversion perspective. J Comp Appl Math 198:378–395

    Article  MathSciNet  MATH  Google Scholar 

  13. Calvetti D, Somersalo E (2005) Statistical compensation of boundary clutter in image deblurring. Inverse Probl 21:1697–1714

    Article  MathSciNet  MATH  Google Scholar 

  14. Calvetti D, Somersalo E (2007) Introduction to Bayesian scientific computing - ten lectures on subjective probability. Springer, Berlin

    Google Scholar 

  15. Calvetti D, Somersalo E (2008) Hypermodels in the Bayesian imaging framework. Inverse Probl 24:034013

    Article  MathSciNet  Google Scholar 

  16. Calvetti D, Hakula H, Pursiainen S, Somersalo E (2009) Conditionally Gaussian hypermodels for cerebral source localization. SIAM J Imaging Sci 2:879–909

    Article  MathSciNet  MATH  Google Scholar 

  17. Cramér H (1946) Mathematical methods in statistics. Princeton University Press, Princeton

    Google Scholar 

  18. De Finetti B (1974) Theory of probability, vol 1. Wiley, New York

    MATH  Google Scholar 

  19. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via EM algorithm. J Roy Stat Soc B 39:1–38

    MathSciNet  MATH  Google Scholar 

  20. Dennis JE, Schnabel RB (1996) Numerical methods for unconstrained optimization and nonlinear equations, SIAM, Philadelphia

    Book  MATH  Google Scholar 

  21. Donatelli M, Martinelli A, Serra-Capizzano S (2006) Improved image deblurring with anti-reflective boundary conditions. Inverse Probl 22:2035–2053

    Article  MathSciNet  MATH  Google Scholar 

  22. Franklin JN (1970) Well-posed stochastic extension of ill-posed linear problem. J Math Anal Appl 31:682–856

    Article  MathSciNet  MATH  Google Scholar 

  23. Fox C, Nicholls G (2001) Exact MAP states and expectations from perfect sampling: Greig, Porteous and Seheult revisited. AIP Conf Proc ISSU 568:252–263

    Article  MathSciNet  Google Scholar 

  24. Gantmacher FR (1990) Matrix theory. AMS, New York

    Google Scholar 

  25. Gelfand AE, Smith AFM (1990) Sampling-based approaches to calculating marginal densities. J Amer Stat Assoc 85:398–409

    Article  MathSciNet  MATH  Google Scholar 

  26. Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions and Bayesian rerstoration of images. IEEE Trans Pattern Anal Mach Intell 6:721–741

    Article  MATH  Google Scholar 

  27. Geyer C (1992) Practical Markov chain Monte Carlo. Stat Sci 7:473–511

    Article  Google Scholar 

  28. Golub G, VanLoan (1996) Matrix computations. Johns Hopkins University Press, London

    MATH  Google Scholar 

  29. Green PJ (1990) Bayesian reconstructions from emission tomography data using modified EM algorithm. IEEE Trans Med Imaging 9:84–93

    Article  Google Scholar 

  30. Green PJ, Mira A (2001) Delayed rejection in reversible jump Metropolis-Hastings. Biometrika 88:1035–1053

    MathSciNet  MATH  Google Scholar 

  31. Haario H, Saksman E, Tamminen J (2001) An adaptive Metropolis Algorithm. Bernoulli 7: 223–242

    Article  MathSciNet  MATH  Google Scholar 

  32. Haario H, Laine M, Mira A, Saksman E (2006) DRAM: Efficient adaptive MCMC. Stat Comput 16:339–354

    Article  MathSciNet  Google Scholar 

  33. Hansen PC (1998) Rank-deficient and ill-posed inverse problems. SIAM, Philadelphia

    Book  Google Scholar 

  34. Hansen PC (2010) Discrete inverse problems. Insights and algorithms. SIAM, Philadelphia

    Google Scholar 

  35. Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109

    Article  MATH  Google Scholar 

  36. Herbert T, Leahy R (1989) A generalized EM algorithm for 3D Bayesian reconstruction from Poisson data using Gibbs priors. IEEE Trans Med Imaging 8:194–202

    Article  Google Scholar 

  37. Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bureau Stand 49:409–436

    MathSciNet  MATH  Google Scholar 

  38. Huttunen JMJ, Kaipio JP (2009) Model reduction in state identification problems with an application to determination of thermal parameters. Appl Num Math 59:877–890

    Article  MathSciNet  MATH  Google Scholar 

  39. Jeffreys H (1946) An invariant form for the prior probability in estimation problem. Proc Roy Soc London A 186:453–461

    Article  MathSciNet  MATH  Google Scholar 

  40. Ji S, Carin L (2007) Bayesian compressive sensing and projection optimization. Proceedings of 24th international conference on machine learning, Cornvallis

    Google Scholar 

  41. Kaipio J, Somersalo E (2004) Statistical and computational inverse problems. Springer, Berlin

    Google Scholar 

  42. Kaipio JP, Somersalo E (2007) Statistical inverse problems: discretization, model reduction and inverse crimes. J Comp Appl Math 198:493–504

    Article  MathSciNet  MATH  Google Scholar 

  43. Kelley T (1999) Iterative methods for optimization. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  44. Laksameethanasan D, Brandt SS, Engelhardt P, Renaud O, Shorte SL (2007) A Bayesian reconstruction method for micro-rotation imaging in light microscopy. Microscopy Res Tech 71: 158–167

    Article  Google Scholar 

  45. Lagendijk RL, Biemond J (1991) Iterative identification and restoration of images. Kluwer, Boston

    Book  MATH  Google Scholar 

  46. LeCam L (1986) Asymptotic methods in statistical decision theory. Springer, New York

    Book  Google Scholar 

  47. Lehikoinen A, Finsterle S, Voutilainen A, Heikkinen LM, Vauhkonen M, Kaipio JP (2007) Approximation errors and truncation of computational domains with application to geophysical tomography. Inverse Probl Imaging 1: 371–389

    Article  MathSciNet  MATH  Google Scholar 

  48. Liu JS (2003) Monte Carlo strategies in scientific computing. Springer, Berlin

    Google Scholar 

  49. Lucy LB (1974) An iterative technique for the rectification of observed distributions. Astron J 79:745–754

    Article  Google Scholar 

  50. Melsa JL, Cohn DL (1978) Decision and estimation theory. McGraw-Hill, New York

    MATH  Google Scholar 

  51. Metropolis N, Rosenbluth AW, Teller AH, Teller E (1953) Equations of state calculations by fast computing machines. J Chem Phys 21: 1087–1092

    Article  Google Scholar 

  52. Mugnier LM, Fusco T, Conan, J-L (2004) Mistral: a myopic edge-preserving image restoration method, with application to astronomical adptive-optics-corrected long-exposure images. J Opt Soc Am A 21:1841–1854

    Article  MathSciNet  Google Scholar 

  53. Nummelin E (2002) MC’s for MCMC’ists. Int Stat Rev 70:215–240

    Article  MATH  Google Scholar 

  54. Ollinger JM, Fessler JA (1997) Positron-emission tomography. IEEE Signal Proc Mag 14:43–55

    Article  Google Scholar 

  55. Paige CC, Saunders MA (1982) LSQR: An algorithm for sparse linear equations and sparse least squares. TOMS 8:43–71

    Article  MathSciNet  MATH  Google Scholar 

  56. Paige CC, Saunders MA (1982) Algorithm 583; LSQR: Sparse linear equations and least-squares problems. TOMS 8:195–209

    Article  MathSciNet  Google Scholar 

  57. Richardson HW (1972) Bayesian-based iterative method of image restoration. J Opt Soc Am 62:55–59

    Article  Google Scholar 

  58. Robert CP, Casella (2004) Monte Carlo statistical methods. Springer, New York

    MATH  Google Scholar 

  59. Saad Y (2003) Iterative methods for sparse linear systems. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  60. Shepp LA, Vardi Y (1982) Maximum likelihood reconstruction in positron emission tomography. IEEE Trans Med Imaging MI-1:113–122

    Google Scholar 

  61. Smith AFM, Roberts RO (1993) Bayesian computation via Gibbs sampler and related Markov chain Monte Carlo methods. J Roy Stat Soc B 55:3–23

    MathSciNet  MATH  Google Scholar 

  62. Snyder DL (1975) Random point processes. Wiley, New York

    MATH  Google Scholar 

  63. Starck JL, Pantin E, Murtagh F (2002) Deconvolution in astronomy: a Review. Publ Astron Soc Pacific 114:1051–1069

    Article  Google Scholar 

  64. Tan SM, Fox C, Nicholls GK, Lecture notes (unpublished), Chap 9, http://www.math.auckland.ac.nz/

  65. Tierney L (1994) Markov chains for exploring posterior distributions. Ann Stat 22:1701–1762

    Article  MathSciNet  MATH  Google Scholar 

  66. Tipping ME (2001) Sparse Bayesian learning and the relevance vector machine. J Mach Learning Res 1:211–244

    MathSciNet  MATH  Google Scholar 

  67. Tipping ME, Faul AC (2003) Fast marginal likelihood maximisation for sparse Bayesian models. Proceedings of the 19th international workshop on artificial intelligence and statistics, Key West, 3–6 January

    Google Scholar 

  68. Van Kempen GMP, Van Vliet LJ, Verveer PJ (1997) A quantitative comparison of image restoration methods in confocal microscopy. J Microscopy 185:354–365

    Article  Google Scholar 

  69. Wei GCG, Tanner MA (1990) A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algoritms. J Amer Stat Aassoc 85:699–704

    Article  Google Scholar 

  70. Wu J (1983) On the convergence properties of the EM algorithm. Ann Stat 11:95–103

    Article  MATH  Google Scholar 

  71. Zhou J, Coatrieux J-L, Bousse A, Shu H, Luo L (2007) A Bayesian MAP-EM algorithm for PET image reconstruction using wavelet transform. Trans Nucl Sci 54:1660—1669

    Article  Google Scholar 

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Authors
  1. Daniela Calvetti
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  2. Erkki Somersalo
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Editor information

Editors and Affiliations

  1. Computational Science Center, University of Vienna, Nordbergstrasse 15, Vienna, Austria

    Otmar Scherzer

  2. RICAM, Austrian Academy of Sciences, Linz, Austria

    Otmar Scherzer

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Calvetti, D., Somersalo, E. (2011). Statistical Methods in Imaging. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92920-0_21

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